On the dimension of the kernel of the Dirichlet problem for fourth-order equations

Abstract

We study the dimension of the kernel of the Dirichlet problem in the disk for fourth-order elliptic equations with constant complex coefficients in general position. Special attention is paid to improperly elliptic equations, because the kernel of the Dirichlet problem for properly elliptic equations is finite-dimensional. We single out classes of improperly elliptic equations that have properties similar to those of properly elliptic equations: the kernel of the Dirichlet problem is also finite-dimensional and in some cases even trivial. We consider fourthorder equations in general position, including equations of principal type as well as equations that have multiple characteristics and whose characteristic equation has roots ±i.

Maximum principle for the weak solutions of the Cauchy problem for the fourth‐order hyperbolic equations

“…We assume, that Equation ( 1) is hyperbolic, that means that all roots of characteristics equation 𝐿(1, 𝜆) = 𝑎 0 𝜆 4 + 𝑎 1 𝜆 3 + 𝑎 2 𝜆 2 + 𝑎 3 𝜆 + 𝑎 4 = 0 are prime, real and are not equal to ±𝑖, that means that the symbol of Equation ( 1) is non-degenerate or that the Equation ( 1) is a principal-type equation. The equations for which the roots of the corresponding characteristic equation are multiple and can take the values ±𝑖 are called the equation with degenerate symbol (see [7]).…”

“…Later, the problem of well‐posedness of boundary‐value problems for various types of second‐order differential equations was studied by Burskii and Zhedanov [2, 3] which developed a method of traces associated with a differential operator and applied this method to establish the Poncelet, Abel, and Goursat problems, and by Kmit and Recke [14]. In the previous works of author (see [6]) there have been developed qualitative methods of studying Cauchy problems and nonstandard in the case of hyperbolic equations Dirichlet and Neumann problems for the linear fourth‐order equations (moreover, for an equation of any even order $$2\,m,\, m\ge 2$$,) with the help of operator methods (L‐traces, theory of extension, moment problem, method of duality equation‐domain and others) [4, 8].…”